1. ## Isomorphism

Define for each $\displaystyle \lambda\in\mathbb{R}$ a natural action on $\displaystyle T_{\alpha}$ by rotation:
$\displaystyle \tau(\lambda)(f)(t) = \left\{ \begin{array}{lr} f(t+\lambda) & : t+\lambda \in [0,1]\\ \alpha^n(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z} \end{array} \right.$
where $\displaystyle \alpha$ is an automorphism of the $\displaystyle C^*$-algebra A. Prove that the one parameter family $\displaystyle \lambda\mapsto \tau(\lambda)$ gives rise to an isomorphism

$\displaystyle T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))$

where $\displaystyle T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}$

2. Originally Posted by Mauritzvdworm
Define for each $\displaystyle \lambda\in\mathbb{R}$ a natural action on $\displaystyle T_{\alpha}$ by rotation:
$\displaystyle \tau(\lambda)(f)(t) = \left\{ \begin{array}{lr} f(t+\lambda) & : t+\lambda \in [0,1]\\ \alpha(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z} \end{array} \right.$
where $\displaystyle \alpha$ is an automorphism of the $\displaystyle C^*$-algebra A. Prove that the one parameter family $\displaystyle \lambda\mapsto \tau(\lambda)$ gives rise to an isomorphism

$\displaystyle T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))$

where $\displaystyle T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}$
I can't help with this problem, which probably requires a fairly extensive knowledge of semidirect products (and in any case I am about to go away for a week). But the problem as stated has two obvious mistakes.

First, the definition of $\displaystyle \tau(\lambda)$ is wrong. It does not define an automorphism group. In fact, the definition of $\displaystyle \tau(\lambda)(f)(t)$ does not even define a continuous function of t at integer values of $\displaystyle t+\lambda$. The correct definition should presumably be $\displaystyle \tau(\lambda)(f)(t) = \alpha^n(f(t+\lambda-n))$ when $\displaystyle t+\lambda\in[n,n+1]$.

Second, the semidirect product $\displaystyle T_\alpha\rtimes_\alpha\mathbb{R}$ does not make sense. It should be $\displaystyle T_\alpha\rtimes_\lambda\mathbb{R}.$